# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | Oct-Nov | (P1-9709/01) | Q#8

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Question

A function f is defined by for .

i.       Find an expression, in terms of , for and show that f is an increasing function.

ii.       Find an expression, in terms of , for and find the domain of .

Solution

i.

We have the function; The expression for represents derivative of .  Rule for differentiation is of is:  Rule for differentiation is of is:  Rule for differentiation is of is:      To test whether a function is increasing or decreasing at a particular point , we  take derivative of a function at that point.

If , the function is increasing.

If , the function is decreasing.

If , the test is inconclusive.

Since is positive for all values of , will be positive, always.

Hence is an increasing function.

ii.

We have; We write it as; To find the inverse of a given function we need to write it in terms of rather than in terms of .          Interchanging ‘x’ with ‘y’;    The set of numbers for which function is defined is called domain of the function.

Finding domain of a function :

·       Since denominator of a fraction cannot be zero, equate denominator of function with zero and solve for set of values of which produce zero in the denominator. Exclude these values  of from domain of ·       Since the number under square root must be positive , make the inequality of number under  square root with and solve for set of values of which produce number under square root.  Exclude these values of from domain of At a glance, we might be tempted to find domain of by considering . That is an  option if were defined for all .

Domain and range of a function become range and domain, respectively, of its inverse function .

Domain of a function Range of Range of a function Domain of Therefore, if we have range of , we can find the domain of .

The function f is defined by for . Therefore, we can find the range of by substituting the extreme values of in it. At ; At ; Hence range of ; Since range of a function is domain of its inverse function, the domain of is; 